So, I completely flunked the Math GRE in September. I studied for it out of the Cracking the GRE book and was doing fairly well. I've done practice exams and done passably well. Then, the test was very different than I expected. I'm registered to take it again in October. I just don't know how to prepare. Advice would be appreciated.

The first step to doing better and developing a plan of action is being more precise and analytical about your situation. What is your mathematical background? What courses have you taken? What does "passably well" mean? What exactly were your scores on the official practice tests? And why, exactly, did you feel the real test was "very different" from the practice tests? In what ways? What content have you not learned?

Unfortunately you don't have much time to prepare for a retake, but you should start by answering the obvious questions. Then we might be able to point you to the most effective resources for improving in a short time frame.

In terms of courses that pop up on the GRE, I've taken calc I-III (of course), linear algebra, multiple analysis courses (real and complex), group theory, and tons of differential equations. I know metric space topology and general notions of open and closed sets, closure, compactness, and connectedness. I've gotten nearly all A's. In terms of major knowledge gaps, I don't know too much about rings and number theory, and I know next to nothing about probability and combinatorics.

I was getting around the 50th percentile, since I can't seem to see the tricks to some of the problems, and I would often get the problems wrong where you pick which of I, II, III are true or any combination of them. It's strange- I seem to get a lot of analysis problems wrong of that type, since I don't have time to prove statements or construct witty counterexamples. I scored much below this, though, although I think that I guessed on too many problems.

I suppose that I didn't expect the types of problems that were on there. Lots of really strange integrals that I couldn't see an obvious way to attack. I felt that the Cracking the GRE problems were much, much easier and required less tricks.

The majority of the integration problems I have seen, on two real tests and all the practice tests, involve integration by parts. Many of the analysis questions involve the mean value theorem or the fundamental theorem of calculus.

As for the construction of counterexamples: I think your performance on this kind of question reflects how deeply you have mastered the art of modeling definitions, not any specific factual knowledge. You should always start by thinking about extreme models: the simplest possible structure that satisfies a definition as well as the wildest or most extreme. In my experience the Roman numeral questions on the GRE test your ability to quickly see this spectrum of possibilities. These are important problem-solving heuristics: look at extreme cases; start as simple as possible; don't build into your model any more complexity than is explicitly required. To do that last bit, you should intentionally pare away any extraneous structure. Don't start with the real numbers, for example, if you could start with a finite set, or with the integers, or with the rationals.

It sounds like you should revisit your performance on the practice tests and think carefully about each problem you missed. Ask yourself: did I miss this because there are basic facts I don't know, or did I miss this because I wasn't able to quickly imagine the right models or the relevant theorems, or did I miss this because I forgot how to calculate? Then the first thing to do is fix the holes in your knowledge of basic facts. For the other sorts of errors, you can drill calculation-heavy problems in calculus or linear algebra textbooks, and you can start building a repertoire of important counterexamples.

There is also some strategy to taking the GRE. With those I, II, III type problems, look at the possible answers first. See which ones you could exclude immediately by checking one of the conditions (for instance, maybe II is true appears in only two of the answers, so you could try checking II first). It may come to the point where you are certain I, III are true (for example), and the only answer with both of those has that IV must be true as well. There's no need to actually check IV then.